Файл: Ufimtsev P. Fundamentals of the physical theory of diffraction (Wiley 2007)(348s) PEo .pdf

Скачать файл (2,80Мб)

Заказать решение

ВУЗ: Не указан

Категория: Не указан

Дисциплина: Не указана

Добавлен: 28.06.2024

Просмотров: 972

Скачиваний: 0

ВНИМАНИЕ! Если данный файл нарушает Ваши авторские права, то обязательно сообщите нам.

7.9 Improved Theory of Elementary Edge Waves 203

of the general expressions (7.163) and (7.164) and the identities Vt = V thp, Ut = U thp, which are valid in the case α = 2π .

To ﬁnd the total ﬁeld scattered by ﬁnite objects, one should also calculate the contribution generated by the uniform component distributed over the ﬁnite elementary

strips (0 ≤ ξ1,2 ≤ l). In the far zone (R								kl2), it is determined by the integrals
	dζ					eikR l
duh1(0) =		ik sin γ0 sin ϑ sin ϕ						0 jh1(0)(ξ1, ϕ0)e−ikξ1 cos β1 dξ1	(7.179)
	4π						R
and
			dζ		eikR			l
	dus1(0) = −			sin γ0				0 js1(0)(ξ1, ϕ0)e−ikξ1 cos β1 dξ1.	(7.180)
			4π		R

Replacing ξ1, β1, ϕ, ϕ0 here with ξ2, β2, α − ϕ, α − ϕ0, one obtains the equations associated with the ﬁeld from strip 2 (0 ≤ ξ2 ≤ l). These integrals are easily calculated in closed form. We show only those results that relate to the grazing incidence (ϕ0 = π ):

(0)

= u0eikζ cos γ0

dζ eikR sin γ0 sin ϑ sin ϕ

)eikl(1−cos β1) − 1* ,

(7.181)

duh1

4π R

1 − cos β1

√

sin γ0

(0)

= u0e−ikζ cos γ0

dζ eikR

ei3π/4

kl(1−cos β1)

dus1

√π

eit dt.

2π

−

cos β1

(7.182)

For the grazing scattering direction (β1 = 0, ϕ = 0), it follows from these equations that

duh1(0) = 0					(7.183)
and
	dζ eikR
dus1(0) = u0e−ikζ cos γ0	dζ eikR		2kl
		sin γ0		ei3π/4.	(7.184)
	2π R	sin γ0	π	ei3π/4.	(7.184)

As was expected, the ﬁeld duh,s(0) is also free from the grazing singularity.

7.9.2Electromagnetic EEWs

This theory is based on the paper by Uﬁmtsev (2006b).

Asymptotic expressions for electromagnetic EEWs established in Section 7.8 possess the grazing singularity. Careful analysis reveals the reason for this singularity. As in

TEAM LinG

204 Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

(0)

the case of acoustic waves, it turns out that the original deﬁnitions of quantities j and

j(1) are not adequate for actual surface currents induced under the grazing incidence (ϕ0 = π or ϕ0 = α − π ). In this case, for instance, the component jn(1), normal to the edge, does not vanish away from the edge, but instead it transforms into a plane wave

there. Also, the component jn(0) includes the absent reﬂected wave. Therefore, one

(0,1)

should modify the original deﬁnitions of j to avoid the grazing singularity.

(0) ≡ hp

With this purpose we introduce a new uniform component j j identical to the electric surface current induced on the illuminated side of the tangential perfectly

(1)

conducting half-plane. A new nonuniform component of the surface current j is the

difference j(1) = j(t) − j hp, where j(t) is the total current induced on the tangential perfectly conducting wedge.

(1)

EEWs Radiated by the Nonuniform Component j

The incident electromagnetic wave (7.129) undergoes diffraction at a scattering per-

(1)

fectly conducting object (Fig. 8.1) and creates there a surface current j . This current radiates diffracted EEWs. Far from the diffraction point ζ (kR 1), these waves can be presented again in the form of Equations (7.130) and (7.131) with new functions

(1) (1)

F and G . These new functions can be found in the same way as described in Section 7.8 and in the papers by Butorin et al. (1987) and Uﬁmtsev (1991), with a simple modiﬁcation based on the replacement (7.159). We provide here only the ﬁnal results.

(1)

The ﬁeld generated by j is described as

(1)

dζ

(1)(ϑ , ϕ)

eikR

and

dH(1)

(1)

/Z0

= [

]

2π E

with

(1)(ϑ , ϕ)

= [

E0t (ζ ) (1)(ϑ , ϕ)

Z0H0t (ζ )

(1)(ϑ , ϕ)

eikφi(ζ ),

]

where

(7.185)

(7.186)

Fϑ(1)(ϑ , ϕ) = [Ut (σ1, ϕ0) − Uthp(σ1, ϕ0)] sin ϑ
+ [Ut (σ2, α − ϕ0) − ε(α − ϕ0)Uthp(σ2, α − ϕ0)] sin ϑ ,	(7.187)
Fϕ(1)(ϑ , ϕ) = 0,	(7.188)

Gϑ(1)(ϑ , ϕ) = [sin γ0 cos ϑ cos ϕ − cos γ0 sin ϑ cos σ1][Vt (σ1, ϕ0) − Vthp(σ1, ϕ0)]

− [sin γ0 cos ϑ cos(α − ϕ) − cos γ0 sin ϑ cos σ2]
× [Vt (σ2, α − ϕ0) − ε(α − ϕ0)Vthp(σ2, α − ϕ0)],	(7.189)

TEAM LinG

7.9 Improved Theory of Elementary Edge Waves 205

and

Gϕ(1)(ϑ , ϕ) = −[Vt (σ1, ϕ0) − Vthp(σ1, ϕ0)] sin ϕ sin γ0

− [Vt (σ2, α − ϕ0) − ε(α − ϕ0)Vthp(σ2, α − ϕ0)] sin(α − ϕ) sin γ0. (7.190)

It is supposed here that 0 < ϕ0 ≤ π .

In the directions ϑ = π − γ0 associated with the diffraction cone, these expres-

sions are simpliﬁed as
Fϑ(1)(π − γ0, ϕ) = −			1	[ f (ϕ, ϕ0, α) − f hp(ϕ, ϕ0)],	Fϕ(1)(ϑ , ϕ) = 0,

		sin γ0
					(7.191)
1					Gϑ(1)(π − γ0, ϕ) = 0,
Gϕ(1)(π − γ0, ϕ) =			[g(ϕ, ϕ0, α) − ghp(ϕ, ϕ0)],
	sin γ0
					(7.192)

with functions f , g and f hp, ghp deﬁned above in Section 7.9.1. Comparison with Equations (7.167) and (7.168) reveals the following relationships between the electromagnetic and acoustic EEWs:

Fϑ(1)	1				Fs(1)(π − γ0, ϕ)
	(π − γ0, ϕ) = −					(7.193)
			sin γ0
and
	1			Fh(1)(π − γ0, ϕ).
Gϕ(1)(π − γ0, ϕ) =						(7.194)
		sin γ0

According to Section 7.9.1 these functions are free from the grazing singularities as well as from the singularities in the directions of the incident and reﬂected rays.

(0) hp

Field Radiated by the Uniform Component j ≡ j

Here we investigate the ﬁeld radiated by the current j induced on the ﬁnite elementary strips (0 ≤ ξ1,2 ≤ l) belonging to the ﬁnite plane faces of a scattering object (Fig. 7.3). In this investigation we apply Cartesian coordinates x, y, z and x , y , z associated with faces 1 and 2, respectively. Axes x and x belong to faces 1 and 2, respectively, and they are parallel to tangents τ1 and τ2. Utilizing the known solution

TEAM LinG

206

Chapter 7 Elementary Acoustic and Electromagnetic Edge Waves

for the half-plane diffraction problem, one ﬁnds the current on strip 1,

jx(0,1) ≡ jxhp,1 = 2H0ze−ikζ cos γ0 eikξ1(cos2 γ0−sin2 γ0 cos ϕ0)

× 1 −

e−iπ/4

∞

√

· √

sin γ0 cos

ϕ0

eit

(7.195)

2 kξ1

jz(,10) ≡ jzhp,1 = −2e−ikζ cos γ0 eikξ1 cos

γ0

sin γ0

× Y0E0z sin ϕ0e−ik

ξ1 sin

γ0 cos ϕ0

e−iπ/4

∞

√

· √

sin γ0 cos

ϕ0

eit

2 kξ1

sin

e−i3π/4

eikξ1 sin2 γ0

sin ϕ0e−ikξ1 sin2 γ0 cos ϕ0

+ sin γ0

−

√2π kξ1

cos ϕ0e−ikξ1 sin

γ0 cos ϕ0

e−iπ/4

∞

+ H0z cos γ0

√

· √

sin γ0 cos

ϕ0

eit

2 kξ1

(7.196)

cos

e−i3π/4

eikξ1 sin2 γ0

cos ϕ0e−ikξ1 sin2 γ0 cos ϕ0

−

+ sin γ0

(0)

√2π kξ1

where Y

1/Z

is the admittance of free space. The current components j

x ,2

and

(0)

0 =

jz,2

on strip 2 are determined by Equations (7.195) and (7.196) with the replacements

H0z → −H0z,

ξ1 → ξ2,

ϕ0 → α − ϕ0.

(7.197)

The ﬁeld radiated by these currents is determined (in terms of the retarded vector-

potential dA) as

dEϕ = −Z0 dHϑ = ikZ0[−dAx,1 sin ϕ + ε(α − ϕ0)dAx ,2 sin(α − ϕ)],

(7.198)

and

dEϑ = Z0 dHϕ = ikZ0{−[dAz,1 + ε(α − ϕ0)dAz,2] sin ϑ

+ [dAx,1 cos ϕ + ε(α − ϕ0)dAx ,2 cos(α − ϕ)] cos ϑ ]}.

(7.199)

It is supposed here that 0 < ϕ0 ≤ π . For the far zone (R

kl 2), one can use the

approximation

dζ

sin γ

eikR

l j

(ξ

)e−ikξ1,2 cos β1,2 dξ

(7.200)

1,2

4π

1,2

TEAM LinG

7.9 Improved Theory of Elementary Edge Waves 207

(0)

After substitution of j 1,2 into Equation (7.200), this leads to the following expressions:

H0ze−ikζ cos γ0

dζ

eikR

dAx,1 = −

[B3(ϕ, ϕ0) − B1(ϕ, ϕ0)] sin γ0

(7.201)

2π

e−ikζ cos γ0

dζ eikR

dAz,1 = −

2π

Y0E0z

B3(ϕ, ϕ0) sin ϕ0

B2(ϕ) sin

B1(ϕ, ϕ0) sin ϕ0

sin γ0

ϕ0

−

B1(ϕ, ϕ0) cos ϕ0 ,

H0z cos γ0 B3(ϕ, ϕ0) cos ϕ0

B2(ϕ) cos

−

sin γ0

(7.202)

where

(ϕ, ϕ

)

eikl (ϕ,ϕ0) − 1

(7.203)

(ϕ, ϕ0)

e−iπ/4

√

kl (ϕ)

B2(ϕ) =

√

eit

dt,

(7.204)

(ϕ)

(ϕ, ϕ0)

ikl (ϕ,ϕ0) e−iπ/4 ∞

(ϕ, ϕ0)

√π

√

2kl

sin γ0 cos

ϕ0

−

+ B2(ϕ) sin γ0 cos

(7.205)

with

(ϕ, ϕ0) = cos2 γ0 − sin2 γ0 cos ϕ0 − cos β1,

(7.206)

(ϕ) = 1 − cos β1.

(7.207)

Components dAx ,2 and dAz,2 are found from dAx,1 and dAz,1, respectively, with

replacements H0z → −H0z, ϕ → α − ϕ, ϕ0 → α − ϕ0, and β1 → β2.

We then substitute the above equations for the vector dA into Equations (7.198) and (7.199) and obtain the ﬁeld expressions in the form of Equations (7.130) and (7.131):

dE(0)

dζ

(0)(ϑ , ϕ)

eikR

(0)

/Z0,

(7.208)

= [

]

(0)(ϑ , ϕ)

2π E